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Csch






Mathematica Notation

Traditional Notation









Elementary Functions > Csch[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving hyperbolic functions > Involving cosh and coth





http://functions.wolfram.com/01.23.21.0162.01









  


  










Input Form





Integrate[(A + B Coth[z] + C Csch[z])/(a + b Cosh[z]), z] == ((Sinh[z] (A + B Coth[z] + C Csch[z]))/(C + B Cosh[z] + A Sinh[z])) (-((2 A ArcTan[((a - b) Tanh[z/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2]) + ((B - C) Log[Cosh[z/2]])/(a - b) + (((-a) B + b C) Log[a + b Cosh[z]])/ (a^2 - b^2) + ((B + C) Log[Sinh[z/2]])/(a + b))










Standard Form





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MathML Form







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</mo> <mrow> <mi> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> C </mi> <mo> &#8290; </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> C </mi> <mo> + </mo> <mrow> <mi> B </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> A </mi> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> A </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; 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</mo> <mi> B </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> B </mi> <mo> + </mo> <mi> C </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <plus /> <ci> A </ci> <apply> <times /> <ci> B </ci> <apply> <coth /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> C </ci> <apply> <csch /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <sinh /> <ci> z </ci> </apply> <apply> <plus /> <ci> A </ci> <apply> <times /> <ci> B </ci> <apply> <coth /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> C </ci> <apply> <csch /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> C </ci> <apply> <times /> <ci> B </ci> <apply> <cosh /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> A </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> A </ci> <apply> <arctan /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <tanh /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> B </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> C </ci> </apply> </apply> <apply> <ln /> <apply> <cosh /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> C </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> B </ci> </apply> </apply> </apply> <apply> <ln /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <cosh /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> B </ci> <ci> C </ci> </apply> <apply> <ln /> <apply> <sinh /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List["A_", "+", RowBox[List["B_", " ", RowBox[List["Coth", "[", "z_", "]"]]]], "+", RowBox[List["C", " ", RowBox[List["Csch", "[", "z_", "]"]]]]]], RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cosh", "[", "z_", "]"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sinh", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["A", "+", RowBox[List["B", " ", RowBox[List["Coth", "[", "z", "]"]]]], "+", RowBox[List["C", " ", RowBox[List["Csch", "[", "z", "]"]]]]]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", "A", " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", RowBox[List["Tanh", "[", FractionBox["z", "2"], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], "+", SuperscriptBox["b", "2"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["B", "-", "C"]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["Cosh", "[", FractionBox["z", "2"], "]"]], "]"]]]], RowBox[List["a", "-", "b"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "B"]], "+", RowBox[List["b", " ", "C"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cosh", "[", "z", "]"]]]]]], "]"]]]], RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["B", "+", "C"]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["Sinh", "[", FractionBox["z", "2"], "]"]], "]"]]]], RowBox[List["a", "+", "b"]]]]], ")"]]]], RowBox[List["C", "+", RowBox[List["B", " ", RowBox[List["Cosh", "[", "z", "]"]]]], "+", RowBox[List["A", " ", RowBox[List["Sinh", "[", "z", "]"]]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2002-12-18





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